Introduction to the Finite Volumes Method. Application to the Shallow Water Equations. Jaime Miguel Fe Marqués

Transcription

1 Introduction to the Finite Volumes Method. Application to the Shallow Water Equations. Jaime Miguel Fe Marqués

2 Contents Preliminary considerations 3. Study of the movement of a fluid Number of dimensions of the model Discretization techniques Systems of hyperbolic equations The Shallow Water Equations One dimensional approach 7. Introduction Conservative variables and conservation laws The Riemann problem Centered and non-centered discretization Numerical diffusion or viscosity Conservative schemes Integral Form Numerical fluxes Convergence Consistency condition Stability condition Conservative scheme Godunov Method for a scalar equation Rankine-Hugoniot jump condition Hyperbolic linear systems Non-centered schemes for linear systems Two-dimensional flow equations 5 3. Types of flow. Turbulent flow Average value and fluctuation Navier-Stokes Equations Reynolds Equations in 3D The Shallow Water equations

3 CONTENTS 4 Application to the D SWE 3 4. Types of finite volumes Cell-type finite volumes Vertex-type finite volumes Edge-type finite volumes Description of the finite volumes used Terms considered in the equations Integration and discretization Discretization of the time derivative Integration of the flux and source terms Definition of the discretized flux Definition of the discretized source Discretization of the boundary conditions Obtaining of the time step Algorithm Some results 4 5. Types of boundary conditions One dimensional problems Straight channel with slope and bottom friction Channel with an obstacle on the bottom Dam break Dam break with reflection Two dimensional problems Partial Dam Break Flow in a fishway References 6

4 Chapter Preliminary considerations. Study of the movement of a fluid The motion of a viscous fluid in three dimensions is described by the Navier- Stokes equations, a system of partial differential equations without analytical solution. When solving these equations numerically we may use different approaches. The most ambitious is the Direct Numerical Simulation that solves all fluid movements. In this case we need to use a mesh size at least as fine as the smallest eddies, which can lead, in a medium size problem, to a number of computing nodes the order of 9. Moreover, the frequency of the fastest events can be about 4 Hz which imposes a time step of not more than 4 s to treat it properly. A second approach, less expensive, is to simulate only the large eddies (Large Eddy Simulation, LES) modeling the effect of those of smaller size, which cannot be solved with a given mesh. The process can be described as filtering equations, after which the velocity field contains only the larger components. This process introduces a stress terms representing the interaction between the two scales of motion and have a dissipative effect. To calculate the effect of these stresses there are different models known in the literature as SGS (subgrid scale models). The LES is based on the work of Smagorinsky [4]. In engineering applications, there is usually no need to know all the details of a flow, but only some properties: the discharge through a channel, the velocity distribution in a section or a substance concentration in a certain volume. For these cases there is a third approach, far less expensive than the other two, that produces sufficiently accurate results: they are the Reynolds equations, which are obtained by averaging in time the Navier-Stokes equations (Reynolds Averaged Navier-Stokes, RANS). The time averaging of the 3

5 CHAPTER. PRELIMINARY CONSIDERATIONS 4 variables produce, similarly to the LES case, some terms called Reynolds stresses. The effect of these stresses can be modeled by estimating a turbulent viscosity, for which a turbulence model is needed.. Number of dimensions of the model These models take into account the motion of fluid particles in the three spatial directions. This is a suitable approach to represent the three-dimensional reality, but the complexity of 3D models can be intractable even for relatively simple domains. On the opposite end are the D models. They are accurate enough for certain phenomena, such as the movement of fluid in a pipe. The free surface, if it exists, is determined by the value of the variable depth (h). The equations are greatly simplified, leading to significant savings in computation time. The problem is that, in most cases, these models do not represent adequately the real problem, not permitting, for instance, taking into account the effect of a change of direction or an asymmetrical section. However, there are a number of phenomena in which fluid motion occurs mainly in two dimensions, for example when the bottom slope is small and the movement of the particles is substantially parallel to it. This makes the D models an interesting option with considerable saving compared to 3D, and they allow an approximation to reality much greater than that achieved with D models..3 Discretization techniques With regard to the spatial discretization, most of the models used in Computational Fluid Dynamics (CFD) use one of the three following techniques: finite differences, finite elements or finite volumes. The Finite Difference Method is the oldest of the three, although its popularity has declined, perhaps due to its lack of flexibility from the geometric point of view. It is usually applied to structured meshes. Its implementation is simple, so new numerical schemes can easily be developed (especially in D) that can be generalized to several dimensions and used in finite volume formulations. The Finite Volume formulation is now widely used in computational fluid dynamics, being its use very common in the field of shallow water equations [3] and 3D models [33]. It is applied to both structured and unstructured meshes with different shapes of the volumes. Its flexibility and conceptual

6 CHAPTER. PRELIMINARY CONSIDERATIONS 5 simplicity explain the acceptance it has. It has been used in commercial programs []. In one dimension it is equivalent to the Finite Difference Method and, depending on the mesh used and the type of discretization, it can also be so with a higher number of dimensions. The main advantage of the Finite Element Method stems from its rigorous mathematical foundation that allows a posteriori error estimation. It is conceptually more difficult than the Finite Volume Method and the physical meaning of the proceedings is less easily seen than in this, although its flexibility to adapt to any geometry is similar. It is used by different authors and applied to commercial programs [6]. Some works [9, 35] compare both methods, showing that the Finite Volume Method shares the theoretical basis of the Finite Element Method, since it is a particular case of the Weighted Residuals Formulation. However, the weighting used in the first (constant volumes in the case of first order approximation) allows to take advantage of some properties of conservation, and the resolution algorithms are posed in a very advantageous way..4 Systems of hyperbolic equations Hyperbolic equations systems have been studied by a number of authors over the last decades. In a first phase, the studies focused on homogeneous systems but, since the 8s, more interest has been put in problems with source term, which have more practical applications. The applications were initially oriented to compressible fluids, achieving significant results in aerodynamics. The strong analogy between the equations of compressible and incompressible flows have permitted to apply similar techniques to the shallow water equations, e.g. Glaister [4] using finite differences, or Vázquez Cendón [3], with finite volumes. Donea and Huerta [] apply the Finite Element Method, in permanent and non-permanent problems, both to compressible and incompressible fluids..5 The Shallow Water Equations The behavior of a viscous fluid is governed by the Navier Stokes equations. These equations were derived in 8 by Claude Navier and, independently, by George Stokes in 845. They form a hyperbolic system of nonlinear conservation laws and, due to their complexity, have no analytical solution. For this reason, the D system of Shallow Water (or Saint Venant) Equations has been obtained from them, by imposing several simplifying assumptions.

7 CHAPTER. PRELIMINARY CONSIDERATIONS 6 These equations describe the behavior of a fluid in shallow areas. Despite the strong assumptions used in their obtaining, the results are very close to reality, even in cases where some of these hypotheses are not fulfilled. Some of the the many problems that can be solved are flow in channels and rivers, tidal flows, sea currents or progression of shockwaves. The one-dimensional version of these equations is commonly used in the study of flow in open channels. Despite its considerable simplicity compared to the Navier Stokes equations, even D Saint Venant equations have no analytical solution and must be solved by approximate methods. The increase, in recent decades, of the computer power has allowed an increasing use of the two-dimensional shallow water equations. Since the 7s of last century, the Finite Element Method has begun to be applied to the shallow water equations: Zienkiewicz [34], and Peraire [] are among the authors who have worked on this line. In parallel to this, the use of the Finite Volume method has grown: see, for instance, the worlks of Vázquez Cendón [3] and Alcrudo and Garcia- Navarro [] among many others. The calculation of the velocity field in a given domain permits the study of many problems of practical interest, such as the sediment transport, the evolution of the salt concentration in an estuary or the dispersal of pollutants.

8 Chapter One dimensional approach. Introduction The aim of this chapter is to show the main aspects of the method in one spatial dimension. First, several commonly used terms are defined and some basic concepts in numerical modeling are introduced or reminded. To describe some of the techniques, simple equations in D are used, such as the transport equation. In order to facilitate the application of the method to the particular case of the shallow water equations, the final chapter defines some terms commonly used in open channels hydraulics.. Conservative variables and conservation laws Conservative variables. There is some freedom to choose the variables that describe the flow to study. One possible choice is to take the primitive or physical variables: the density ρ, the pressure p and the three components of velocity, u, v, w. Another one is to use the so-called conservative variables, which result from applying the fundamental laws of conservation (of mass, momentum, energy). These variables are, for example: the three components of momentum per unit volume ρu, ρv, ρw and the total energy per unit volume. For systems of equations governing the free surface flow in one or two dimensions, such as the Shallow Water system, the conservative variables commonly used are the depth h and its product by the velocity components: hu in one dimension and (hu, hv) in the two-dimensional case. Conservation laws. They are systems of partial differential equations expressing conservation of m quantities u,... u m. If obtained from a control 7

9 CHAPTER. ONE DIMENSIONAL APPROACH 8 volume fixed in space, which is crossed by the moving fluid, they are said to be written in conservative form, this is the way that most resembles a flow balance of mass and momentum [7, pg. 9]. If the control volume moves with the fluid, so always contains the same particles, the non conservative form is obtained [4, pg. 6]. A conservation law in conservative form is written U t + F x =, U = U(x, t), F = F(U). (.) U is called Variables Vector and F(U) Flux Functions Vector. When expressing the conservation laws in differential form, it is assumed that the solutions satisfy the relevant requirements of regularity. Example (scalar): The Transport equation (linear advection equation). u t + f x = u = u(x, t), f(u) = au, a = constant. (.) Example (scalar): Burgers equation. u t + f x = u = u(x, t), f(u) = u. (.3) Example 3 (system of conservation laws): the one-dimensional shallow water equations. { } { } h hu U =, F(U) = hu hu +. (.4) gh Nonconservative Form. If, for instance, we replace f by its value in the Burgers Equation we obtain the nonconservative form u t + u u x = u = u(x, t). (.5) If instead of an equation we consider a system of conservation laws, applying the chain rule we obtain the following expression: where A is the Jacobian matrix U t + df U du x = U t + A U x = (.6) A = df du = f f... u u m..... f m f m u u m. (.7)

10 CHAPTER. ONE DIMENSIONAL APPROACH 9 The nonconservative formulation (.6) is equivalent to the conservative one (.), and has the same solution, provided that this solution is sufficiently regular. Otherwise the derivation which led to (.6) is not valid. For example, if the solution is discontinuous e.g. a shock wave erroneous results are obtained. Integral Form. The conservation laws can also be expressed in integral form. One reason for the use of this form is that the obtaining of the equations is based on physical conservation principles, generally expressed as integral relationships. On the other hand the integral formulation requires less derivability conditions on solutions, which allows to obtain discontinuous solutions. These discontinuous solutions do not verify the partial differential equation at every point because the derivatives are not defined at the discontinuities, and must meet a jump condition along them, which is obtained from the integral form (see.6.8, Rankine-Hugoniot condition). The solutions of the integral form are known as weak solutions..3 The Riemann problem We will analyze the Riemann problem for the importance it has on the Godunov method, from which the method that is described here derives. The transport equation, already mentioned, has the form u t + a u = u = u(x, t), a = constante. (.8) x The Cauchy problem, applied to this equation consist in solving it with the initial condition u(x, ) = u (x). (.9) As it can easily be seen by substituting in the equation, the solution is given by u(x, t) = u (x at), x R, t, (.) which can be interpreted saying that the function u moves in time, along the axis x, speed a without deforming. The points of the plane x, t in which the above said occurs are called characteristic curves. Their equation in this case is given by dx dt = a ( in general, dx dt = f (u) ) (.) and in them the solution u of the equation remains constant. Indeed, if u is a solution of the equation, the total derivative satisfies du dt = u t + u x dx dt = u t + a u x =. (.)

11 CHAPTER. ONE DIMENSIONAL APPROACH The Riemann problem is a particular case of the former characterized by the initial condition { ul si x <, u(x, ) = u (x) = (.3) u R si x >. The initial discontinuity at x = is propagated to a distance d = at at time t. The characteristic curve x = at separates -in the plane x, t- the points in which the solution is u L from those in which is u R and it is represented in figure. in bold. The Riemann-problem that can be expressed briefly as RP (U L, U R )- has as solution u(x, t) = u (x at) = { ul si x at <, u R si x at >. (.4) If, for example, a >, the wavefront will move to the right. The solution of the equation is u L at all points that have already been reached by the wave, which are those to the left of the point x = at after time t (Figure.). The characteristic curves thus represent the pairs (x, t) corresponding to the d=at ul ul ur ur x= x= Figure.: The Riemann Problem after time t, a >. advance of the wave. For example, at t = t and x = x ul ul the variable takes a t ur ur certain value. To know the position of the point where the variable initially took the same value, we go x=down the curve passing through x= (x, p ) to find the horizontal axis (Figure.). The result, unsurprisingly, is x = x at. ( x*,t* ) t x<at x>at x*-at* ( x*,t* ) x*-at* x d=at x Figure.: Caracteristic curves.

12 CHAPTER. ONE DIMENSIONAL APPROACH.4 Centered and non-centered discretization Before describing the finite volume method (section.6) and applying it to the shallow water equations (chapter 4), we apply the centered and noncentered discretization to the Transport Equation. Equation (.8) is considered again u t + a u x = u = u(x, t), a = constant. (.5) Different discretizations for this equation can be obtained from the Taylor series expansion. For example, if i is the spatial index and n the time index, u n+ i = u n i + u t n i t + u n t i and the time derivative can be approximated as u t n i = un+ i u n i t t + O( t 3 ) (.6) u n t t + O( t ), (.7) i which is a first order forward discretization. Also u n i+ = u n i + u n x + u n x i x x + 3 u n i 6 x 3 x 3 + O( x 4 ), (.8) i u n i = u n i u n x + u n x i x x 3 u n i 6 x 3 x 3 + O( x 4 ), (.9) i and subtracting these two equations, we obtain a space centered second order discretization u n = un i+ u n i 3 u n x i x 6 x 3 x + O( x 3 ). (.) i Then equation (.5) can be written in discretized form as u n+ i u n i t + a un i+ u n i x = (.) from which the following numerical algorithm results u n+ i = u n i a t x (un i+ u n i ). (.) This scheme, first order in time and second in space is called Euler explicit scheme and it can be shown that is unconditionally unstable [7, pg. 63].

13 CHAPTER. ONE DIMENSIONAL APPROACH In order to remedy the lack of stability observed in the above scheme, we discretize spatially in non-centered form, which produces two options u n x i u n x i = un i u n i, (.3) x = un i+ u n i x, (.4) from which only one is successful, depending on the sign of the speed a of the wave. If a >, the option (.3) together with (.7) results u n+ i u n i t + a un i u n i x = (.5) from where being u n+ i = u n i c(u n i u n i ), (.6) c = a t x, (.7) This scheme proves to be stable [7, pg. 64] provided that c. (.8) The parameter c is called the Courant number or the CFL (Courant- Friedrichs-Lewy) number. It can be considered as the ratio of two lengths: the one traveled by the wave in time t and the mesh size x. As a is a datum and x is usually determined by the desired degree of accuracy, one can only vary t to satisfy the stability condition. The scheme (.6) is known as the first order upwind scheme and also the CIR scheme, (Courant, Isaacson and Rees). The name upwind refers to the fact that in the spatial discretization we use grid points from the side where information comes. The CIR has the disadvantage, common to all first order methods of being very diffusive: it tends to smooth discontinuities in the solution and cut extreme values. If, for a >, we introduce (.7) and (.4) in the transport equation (.5), the resulting downwind scheme u n+ i = u n i c(u n i+ u n i ), (.9) is unconditionally unstable. That is, to obtain a useful non-centered scheme the sign of a in the spatial discretization must be taken into account.

14 CHAPTER. ONE DIMENSIONAL APPROACH 3 Lax-Friedrichs, characterized for re- Another first order scheme is the placing the term u n i in (.) by (un i + u n i+), (.3) i.e. the average of the values in the two neighboring nodes. The resulting scheme u n+ i = ( + c)un i + ( c)un i+ (.3) is stable under the condition (.8) [7, pg. 68]..5 Numerical diffusion or viscosity We will try to clarify below the previous section assertion that the first order schemes are diffusive. Let us consider Equation (.8) without time derivative a u x =, a >. (.3) If the non-centered discretization (.3) is applied, we get a u = a u i u i + O( x). (.33) x i x Moreover, from the Taylor series expansion (.9), u i = u i u x + u i x i x x + O( x 3 ), (.34) whence, multiplying by a and rearranging, we obtain That is, the expression a u u i x i a x x = a u i u i + O( x ). (.35) x a u i u i, (.36) x which represents a first order discretization of (.3), is simultaneously a second order discretization (thus more accurate) of a u x a x u =, (.37) x containing a diffusive term with a coefficient a x/.

15 CHAPTER. ONE DIMENSIONAL APPROACH 4 So when discretizing upwind Equation (.3) a so-called numerical diffusion is being introduced. The coefficient that quantifies this diffusion (also called numerical viscosity) depends on the mesh size, so if x is sufficiently small, thereby increasing the computation time, the diffusive effect tends to disappear. If, however, the diffusive effect is high, the extreme values of the solution tend to cut and discontinuities to spread. Another solution to reduce the diffusive effect is to use higher order schemes. These schemes take into consideration the values in a larger number of nodes, so the programming is more complicated. Lowering the numerical viscosity also reduces stability..6 Conservative schemes. The finite volume method in one spatial dimension is based on dividing the spatial domain into intervals (called finite volumes or cells) making in each of them an approximation of the integral of the conservative variables. At each time step these values are updated using approximations of the flux at the ends of the intervals, as it will be discussed below, using the scalar conservation law u t + f x =, u = u(x, t), f = f(u), (.38) that represents the transport equation if f(u) = au, being a a constant..6. Integral Form A way to discretize (.38), considering weak solutions, is to divide the spatial domain into finite volumes and integrate the equation in each cell, transforming it into an integral form. For simplicity, we will use intervals (finite volumes) with equal length x and take a constant time step t. Thus the spatial and temporal domains will be [ I i = x i, x i+ ] = [ x i x, x i + x ], I n = [t n, t n+ ] = [n t, (n + ) t] x i (.39) and the integral in the cell, of Equation (.38) xi+ [ u t + f ] dx =, (.4) x

16 CHAPTER. ONE DIMENSIONAL APPROACH 5 becomes xi+ x i u t dx + f(u(x i+, t)) f(u(x i, t)) =. (.4) Since the interval ends x i± xi+ t x i u(x, t)dx + f(u(x i+ do not depend on time, we can write, t)) f(u(x i, t)) =. (.4) We define u n i as the spatial average of the function u(x, t) in the interval I i, at time t n = n t, i.e. u n i = xi+ u(x, t n )dx, (.43) x x i Integrating (.4) between t n and t n+, the time derivative disappears from the first term, resulting xi+ [u(x, t n+ ) u(x, t n )] dx x i + tn+ t n [ ] f(u(x i+, t)) f(u(x i, t)) dt = (.44) and we see that the value of u in I i only changes along time t due to the value of the flux f at the ends of I i. Then, using (.43), ( u n+ i ) tn+ [ u n i x + t n f(u(x i+ ], t)) f(u(x i, t)) dt =. (.45).6. Numerical fluxes In the above expression, the values of the integral of f at points x i± not be generally known, so we replace them with will so we get f n i± u n+ i t tn+ f(u(x i±, t))dt, (.46) t n = u n i t ( ) f n f n. (.47) i+ x i The explicit algorithm (.47) allows us to obtain the approximation of u in each cell, at time t n+, from its value in the previous time and the numerical fluxes f n at the ends of the cell. i±

17 CHAPTER. ONE DIMENSIONAL APPROACH 6 These numerical fluxes represent approximations of the time average of the physical flux at the edges of the cell and, depending on the way they are calculated, we get different schemes. To calculate them, the variables in cells adjacent to I i are used f n i± = φ(u n i m, u n i m+,..., u n i+l), (.48) where m and l are two non-negative integers and φ is a certain function. In hyperbolic problems information propagates at a finite speed, so it seems reasonable to assume that we can obtain f n from u n i i and u n i (the average values of the variable on both sides of the boundary x i ), while f i+ is obtained from un n i and u n i+. Then the general expression (.48) takes the form f n = φ(u n i i, u n i ) ; f n = φ(u n i+ i, u n i+). (.49).6.3 Convergence The algorithm (.47) allows us to obtain variable values forward in time. To provide a good approximation of the law of conservation, the algorithm must be convergent. which means that the numerical solution converges to the solution of the differential equation when x, t. Convergence is ensured with two requirements: consistency and stability. Indeed, Lax Theorem states that a consistent and stable scheme is convergent. We will briefly discuss both..6.4 Consistency condition We say that a scheme is consistent if it represents faithfully the differential equation when t, x. As we are getting the numerical flux from the values of u in neighboring cells, if u has the same value in all of them, the result must be the same in each one. Therefore, a consistency condition required to function φ is: φ(v, v,..., v) = f(v); (.5) Usually, continuity for the variable u is also required, i.e. φ(u n i u n i ) f(v), when u n i, u n i v [, pg. 68].

18 CHAPTER. ONE DIMENSIONAL APPROACH Stability condition A method must be stable in the sense that a small error introduced at any time step is not amplified indefinitely but remains bounded along the process. In paragraph.4 it was said that the Euler explicit scheme was unconditionally unstable, while the CIR scheme was stable when c. These statements are based on the stability criterion of Von Neumann, which is based on Fourier analysis and is very useful in the study of linear systems..6.6 Conservative scheme A conservative scheme for the scalar conservation law (.38) is a numerical method of the form (.47) that fulfills the condition (.48). We see that by applying a conservative scheme to a set of contiguous cells N, N +... M, the result verifies the same property (.44) of the exact solution (the value of u in I i only changes in time t due to the value of the flow f at the ends of I i ). Indeed, adding the values of u n+ i obtained from (.47), for any set of consecutive cells, multiplying by x and rearranging, we get ( M i=n u n+ i M i=n u n i ) x + ( ) f n f n t =, (.5) M+ N since fluxes at cell boundaries cancel each other, except for flows at the ends x = x N and x = x M+. The interest of conservative schemes is that, as the Theorem of Lax- Wendroff [5, pg. 68] says, if a consistent conservative scheme converges, the result is a weak solution of the equation. In contrast, non-conservative schemes may not converge to the correct solution, if a shock wave is present [7, pg. 7]. Two examples of conservative schemes are the Godunov and Lax-Friedrichs schemes. We may say that the Lax-Wendroff theorem continues Lax s Theorem. That is, a scheme consistent, stable and conservative converges at a weak solution of the equation. The algorithm (.47) can also be seen as a finite difference approximation of the conservation law (.38), as this law can be discretized as u n+ i u n i t + f n i+ f n i x =, (.5) where f n, f n i+ i are approximations of the value of f at the endpoints.

19 CHAPTER. ONE DIMENSIONAL APPROACH Godunov Method for a scalar equation Godunov conducted the first conservative extension of the CIR scheme to nonlinear systems of conservation laws. The Godunov first order upwind method is a conservative scheme in the form (.47), where the numerical fluxes at the boundaries of the cells, f i±, are calculated using solutions of local Riemann problems. It means that a Riemann Problem is solved in every time step at every boundary between two cells, taking as initial values at each side of the boundary, the average values of the variable in the previous time step, as discussed below. It is assumed that in each time step t n, variable u is piecewise constant, taking on each cell I i the value given by (.43). There are, then, a pair of steady states at each boundary of I i : (u i, u i ) on the left and (u i, u i+ ) on the right, both of which can be considered as a local Riemann Problem, originating at x =, t =. Thus, in the left side, x = x i, we have u + f t x =, u(x, ) = u (x) = and, on the right side, x = x i+, { u n i, x <, u n i, x >, (.53) u + f t x =, u(x, ) = u (x) = { u n i, x <, u n i+, x >. (.54) Let ũ(x, t) be the combined solution of RP (u n i, u n i ) and RP (u n i, u n i+) in I i. Since ũ(x, t) is the exact solution of the conservation law (.38), we introduce it in the integral form (.44), with spatial and temporal domains, respectively obtaining I i = [x i, x i+ ], I n = [, t], (.55) xi+ x i ũ(x, t)dx = xi+ x i t ũ(x, )dx f(ũ(x i+, t))dt + t f(ũ(x i, t))dt =. (.56)

20 CHAPTER. ONE DIMENSIONAL APPROACH 9 Now we define, as in (.43) u n i = x xi+ x i ũ(x, )dx, u n+ i = x xi+ ũ(x, t)dx, x i (.57) and it results the conservative scheme (.47) being u n+ i f i± = u n i t ( ) f n f n, (.58) i+ x i = t t f(ũ(x i±, t))dt. (.59) The integrand in (.59) depends on the exact solution to the Riemann Problem, at each end of the cell, along the time axis (in local coordinates, then at x = ). This is represented as whereby ũ(x i±, t) = u i± (), (.6) f i± If, for instance, the flux function is f(u) = au, f i = f(u i± ()). (.6) = au n i, f i+ a >, it results = au n i (.6) (if, instead, we take a < the result is au n i on the left boundary and au n i+ on the right). Replacing in (.58), we arrive at u n+ i = u n i + t x (aun i au n i ) = u n i a t x (un i u n i ), (.63) i.e. the CIR scheme (.6). Thus, Godunov Method considers the problem to be solved as a succession of states, constant in each finite volume. At each time step a Riemann problem at the boundary of each cell is solved, taking the exact solutions of each local problem as the fluxes in these boundaries. These exact solutions must be calculated according to the equation in question, if it is not linear; in [7, pg.76], the exact solutions of Riemann problem in the

21 CHAPTER. ONE DIMENSIONAL APPROACH case of the quasi-linear Burgers equation may be seen. Finally, the spatial averaging of the dependent variables in each cell is performed. To simplify the process, different authors [6, 3, 3] have used schemes called approximate Riemann solvers, which they have applied to compressible fluids. These schemes have been extended later [, 4, 9, 3] to free surface flows with very good results (in [] different approximate Riemann solvers are described). While it is true that these schemes will replace the exact solution of the Riemann problem by an approximate one, the information provided by the exact solution is partially lost in any case, due to spatial averaging in each cell, which makes less significant the error in the approximation [8]. In the conservative scheme of Lax-Friedrichs the fluxes at the ends are calculated as f i+ = + c c f(un i ) + c c f(un i+), (.64) f i = + c c f(un i ) + c c f(un i ), where c takes the value given by (.7). If f(u) = au and we replace (.64) in equation (.47), the finite differences scheme (.3) of Lax-Friedrichs is obtained..6.8 Rankine-Hugoniot jump condition In the preceding description of Godunov s Method, the integral form is discretized, looking for weak solutions to the differential equation (.38), when the initial condition is discontinuous (Riemann problem). Of course, any function u that is a classical solution (hence differentiable) of the equation will be a weak solution. And a weak solution is a classical solution in the intervals in which it is differentiable. In the event that there is a discontinuity in a weak solution u(x, t), the function will take values u L and u R at both sides of the discontinuity. Then the following relationship, known as the Rankine-Hugoniot condition, is verified, (u R u L )S = f(u R ) f(u L ) (.65) being S the speed at which the jump is transmitted. The Rankine-Hugoniot condition is shown in [7, pg. 7], using the Leibnitz formula and in [, pg. ] based on geometrical considerations. In the following examples we obtain this velocity S in two cases. Example. The transport equation, where f(u) = au, a constant. S = f(u R) f(u L ) u R u L = au R au L u R u L = a. (.66)

22 CHAPTER. ONE DIMENSIONAL APPROACH As we already knew (.3), the wave moves at a constant speed a. Example. Burgers equation. Now f(u) = u. S = u R u L u R u L = u R + u L. (.67) In this case, the speed of advance of the wave depends on the values of u at both sides of the discontinuity..7 Hyperbolic linear systems In paragraphs.4 to.6 we have referred to the case of a partial differential equation. Conservation laws are usually given in the form of a system of nonlinear equations. We will describe some techniques, for the case of linear systems, which can be extended to nonlinear systems. Let a linear system of partial differential equations U t + AU x =, U = {u j }, j =,... m, (.68) where U is the variables vector and A m m a constant matrix. The system is called hyperbolic if A is diagonalizable, i.e. if it has m real eigenvalues λ i and m eigenvectors linearly independent k i. It is strictly hyperbolic if all the eigenvalues are different. Calling Λ the diagonal matrix of eigenvalues and X the matrix whose columns are the eigenvectors k i, it holds A = XΛX. (.69) The existence of X allows us to define a new vector of variables V = {v j } j=,...m = X U. (.7) Using the relationships (.69) and U = XV it results, from (.68), XV t + XΛX XV x = XV t + XΛV x = X (V t + ΛV x ) =, (.7) from where V t + ΛV x =. (.7) Thus the system in canonical or characteristic form has been obtained. Each of the resulting m uncoupled equations have only one variable involved v j t + λ v j j x = j =,... m, (.73)

23 CHAPTER. ONE DIMENSIONAL APPROACH and take the form of the D transport equation. Thus, the system can be seen as a combination of m waves traveling at their characteristic velocities given by the m eigenvalues λ j. These eigenvalues (or characteristic values) define the characteristic curves x(t) = x + λ j t, (see.3, along which the information corresponding to each one of the characteristic variables {v j } propagates. With the above said we have made a basis change to the set of eigenvectors k i. The characteristic variables can be interpreted then as the components of the variables vector U in the reference system formed by the eigenvectors (being X the basis change matrix) or, what is the same, as the projections of the vector U on the eigenvectors. After solving the system (.73), the values U can be obtained from the relationship (.7). For example, let us consider the system { } { } u u U t + F x =, U =, F = (.74) u u + u which, by applying the rule of the chain, changes to { } { u U t + AU x =, U =, A = From matrix A we obtain λ =, λ =, X = { u } {, X = }. (.75) }. (.76) Then, using the base change (.7), the characteristic variables are found to be { } { } { } { } v u u = =, v u u u (.77) and, using these variables, we obtain the decoupled system v t + v x = (.78) v t + v x =. (.79) Then, if the initial condition of the problem is the system solution will be, as in (.) v (x, ) = v (x); v (x, ) = v (x), (.8) v (x, t) = v (x t), v (x, t) = v (x t), x R, t (.8) and we can undo the variables change by using the matrix X { } { } { } { } u v v = =. (.8) u v v + v

24 CHAPTER. ONE DIMENSIONAL APPROACH 3.8 Non-centered schemes for linear systems As we have just seen, the linear hyperbolic system (.68) U t + AU x = can decouple, becoming m equations (.73), each one involving a single variable. But the non-centered conservative scheme studied in the scalar case (.38) can only be applied to this system if all eigenvalues have the same sign. Then, in a general case, with eigenvalues of both signs, we will discretize upwind for the positive ones and downwind for the negatives. The practical way to accomplish this is to decompose the matrix A into two, one with positive or zero eigenvalues and the other with negative or zero eigenvalues. Scheme CIR. Calling { λ + λj si λ j = j, si λ j <, { λ λj si λ j = j, si λ j >, (.83) (.84) the CIR scheme (.6) applied to a decoupled hyperbolic linear system (in canonical form) is {v j } n+ i = {v j } n i t x λ+ j ( {vj } n i {v j } n i ) t j =,... m. x λ j ( ) {vj } n i+ {v j } n i, (.85) Let Λ be the diagonal matrix of eigenvalues λ j, Λ + of the λ + j, Λ of the λ j and Λ of the absolute values λ j. It holds If we define it results Λ = Λ + + Λ, (.86) Λ = Λ + Λ. (.87) A + = XΛ + X, (.88) A = XΛ X, (.89) A = X Λ X, (.9) A + + A = A, (.9) A + A = X Λ X = A. (.9)

25 CHAPTER. ONE DIMENSIONAL APPROACH 4 And the CIR scheme can be written then in vector form, either in terms of characteristic variables V n+ i = V n i t x Λ+ (V n i V n i ) t x Λ (V n i+ V n i ), (.93) or -by means of the matrix X - in terms of the starting variables U n+ i = U n i t x A+ (U n i U n i ) t x A (U n i+ U n i ). (.94) Techniques of flux splitting use expressions like (.94). Godunov Method. The linear hyperbolic system (.68), is considered again, now written in conservative form U t + F x =, F(U) = AU. (.95) The Godunov first order upwind scheme uses the conservative formula analogous to (.58) U n+ i = U n i t ( ) F n F n, (.96) i+ x i where, similarly to (.6), the flux terms of the cell borders are ( ) = F (), (.97) being U i F i± U i± () and U i+ () the solutions of RP (U n i, U n i ) and RP (U n i, U n i+) respectively. Numerical flows at the ends of the interval are calculated from the values of F and U in the anterior and posterior points yielding the expressions [7, pg. 85] F i+ F i = ( F n i + Fi+) n A ( ) U n i+ U n i, (.98) = ( ) F n i + F n i A ( ) U n i U n i, (.99) where A takes the value given by (.9). That is, the resulting value for the flux vector at the left and right borders of the cell is the mean of the values at the points (i, i) and (i, i + ) respectively, with an upwinding term. The Q- schemes use expressions like (.98) and (.99).

26 Chapter 3 Two-dimensional flow equations 3. Types of flow. Turbulent flow The importance of the inertia forces with respect to the viscous ones in a particular flow is quantified by the dimensionless Reynolds number, which is calculated as the quotient between the two forces. Considering the magnitudes involved in both forces one obtains the usual expression of Re Re = ρv L µv L = V L ν. (3.) V and L are the characteristic velocity and length of the flow and ν is the ratio between the dynamic viscosity and the density, called kinematic viscosity. It is observed experimentally that for values below the so-called critical Reynolds number the adjacent fluid layers slide over each other in an orderly way, which is called laminar regime. In a laminar regime, if the boundary conditions do not vary with time, the flux is permanent. For values above critical Re, the flow behavior changes, becoming random and chaotic. The movement becomes non permanent, even with constant boundary conditions. It is called turbulent regime. The random nature of turbulent flow and the high frequency with which the magnitudes vary make extremely difficult in practice a complete description of the movement of all fluid particles. Let u, v, w the velocity components and p the pressure. One can decompose a magnitude (for instance a velocity component u) in the sum of its average value (u) and the turbulent fluctuation (u ). A turbulent flow is then characterized by the average values (u, v, w, p) and the statistical properties of the fluctuations (u, v, w, p ). Even in flows where average velocities and pressures vary only in one or two spatial dimensions, the turbulent fluctuations are three-dimensional. If 5

27 CHAPTER 3. TWO-DIMENSIONAL FLOW EQUATIONS 6 we can visualize a turbulent flow we find fluid portions in rotation, called turbulent eddies. These have a wide spectrum of sizes being the largest eddies comparable to the dimensions of the domain. Inertial forces dominate in larger eddies, while its effect is negligible compared with the viscous forces in the smallest. The energy required to maintain the motion of the larger eddies flow comes from the mean flow. On the other hand, smaller eddies obtained energy mainly from the higher ones. Thus kinetic energy is transmitted to increasingly smaller eddies through a cascade process, until it is dissipated by viscous forces. This dissipation causes the additional energy losses related to the turbulent flows. 3. Average value and fluctuation As mentioned in the previous section, a magnitude ϕ, which generally depends on time, can be decomposed into the sum of its average value plus a fluctuation component around this value. ϕ(t) = ϕ(t) + ϕ (t). (3.) Although it does not appear explicitly in the expression of ϕ and ϕ, both are a function of the coordinates of the considered point (x, y, z). The temporal average of ϕ(t), for a given point, can be defined in different ways. For the cases considered here, of unsteady flow we will use the expression [, pg. 78] ϕ(t) = t t+ t t t ϕ(θ)dθ, (3.3) where the choosen t must be greater than the time scale of the turbulence and lower than the time scale of the average flow. For example, in an estuary it can be considered that the period of the turbulent oscillation of the velocity is less than one second, while the tide period is about hours. After performing the time average (3.3), the mean flow continues oscillating under the influence of the tide. 3.3 Navier-Stokes Equations The shallow water equations are derived from the Navier-Stokes equations, which govern the behavior of a viscous fluid in three dimensions.

28 CHAPTER 3. TWO-DIMENSIONAL FLOW EQUATIONS 7 In incompressible fluids, density is independent of the pressure. In these fluids, mass per unit of volume can vary, for example due to temperature, but is considered constant with respect to the position and time. Be a Cartesian system x, y, z, with z positive upward. Calling u, v, w the components of the velocity vector u, these equations are expressed as: Continuity Equation (conservation of mass) u x + v y + w z Dynamic Equation (conservation of momentum) =. (3.4) du dt dv dt dw dt = u t + u u x + v u y + w u = v t + u v x + v v y + w v = w t + u w x + v w y + w w z = F x p + ν u, ρ x (3.5) z = F y p + ν v, ρ y (3.6) z = F z p + ν w. ρ z (3.7) The vector F = (F x, F y, F z ) T is the force per unit mass; p is the pressure; ρ is the density; ν = µ ρ is the kinematic viscosity, µ is the dynamic viscosity. Adding to each of the three dynamic equations the continuity equation multiplied by u, v and w respectively, and using the operator for divergence and for the Laplacian, the system takes the form u =, (3.8) u t + uu = F x p ρ x + ν u, (3.9) v t + vu = F y p ρ y + ν v, (3.) w t + wu = F z p ρ z + ν w. (3.) 3.4 Reynolds Equations in 3D These equations govern turbulent movement of an incompressible fluid. To get them we replace, in the Navier-Stokes equations, the velocity and pressure by their time-averaged values plus the fluctuation terms. u = (u, v, w), u = u + u, v = v + v, w = w + w, p = p + p (3.)

29 CHAPTER 3. TWO-DIMENSIONAL FLOW EQUATIONS 8 and calculate the time average of each equation, obtaining u =, (3.3) u t + u u + u u = F x p ρ x + ν u, (3.4) v t + v u + v u = F y p ρ y + ν v, (3.5) w t + w u + w u = F z p ρ z + ν w. (3.6) The resulting equations have the same structure as equations (3.8)-(3.), with two differences: the variables u, v, w, p have been replaced by their average values and new ones have been added on the left member. If we develop them and place them on the right, we obtain the 3D Reynolds equations: u =, (3.7) [ ] u t + u u = F x p ρ x + u ν u x + u v y + u w, (3.8) z [ ] v t + v u = F y p ρ y + u ν v v x + v y + v w, (3.9) z [ ] w t + w u = F z p ρ z + u ν w w x + v w y + w.(3.) z The cross products of the turbulent fluctuations of velocity multiplied by density have dimensions of force/area and they are called Reynolds stresses. According to the Boussinesq hypothesis, these stresses are proportional to the derivative of the time averages of the velocity components, being µ t, turbulent dynamic viscosity, the coefficient of proportionality. Operating, simplifying and assuming ν + ν t ν t we obtain: u =, (3.) u t + u u = F x p ρ x + u ν t u + ν t x, (3.) v t + v u = F y p ρ y + u ν t v + ν t y, (3.3) w t + w u = F z p ρ z + u ν t w + ν t z. (3.4) These expressions are very similar to the Navier Stokes Eq. (3.8)-(3.) with the difference that the instantaneous values of velocity and pressure have

30 CHAPTER 3. TWO-DIMENSIONAL FLOW EQUATIONS 9 been replaced by their time averages, the viscosity by the turbulent viscosity ν t = µ t /ρ and a new addend has appeared in the source term, which is negligible if we assume that the spatial variation of ν t is very small. 3.5 The Shallow Water equations There are flows in nature in which the horizontal dimensions are clearly predominant. If, in addition, the vertical variation in the horizontal velocity component is small and there are little vertical accelerations, these flows can often be described by means of a set of equations in two dimensions, resulting of the vertical integration of the 3D equations. To obtain them some hypothesis are made: a) Small bottom slope. b) The distribution of pressure is assumed to be hydrostatic. c) The main movement of particles occurs in horizontal planes. d) The vertical distribution of u, v is nearly uniform. e) The mass forces are gravity and the Coriolis force. f) The vertical acceleration of the particles is negligible compared to g. g) The contours friction in unsteady flow, can be evaluated using formulae valid for steady flow, like Manning. The shallow water equations in two dimensions are expressed as U t + F x + F y = G, (3.5) being the variables vector U = h hu hv, (3.6) terms F and F F = hu hu + gh huv, F = hv huv hv + gh, (3.7)

31 CHAPTER 3. TWO-DIMENSIONAL FLOW EQUATIONS 3 and the source term G = fvh + τ s x ρ + gh(s x S fx ) + S t fuh + τ s y ρ + gh(s y S fy ) + S t. (3.8) The variable h represents the depth measured vertically, u, v are the averages in the vertical of the horizontal components of the velocity, f is the Coriolis coefficient, τ s evaluates the effect of wind, S and S F are the geometric and frictional slopes. S x = z b x, S y = z b y, (3.9) S fx = n u u + v, S h 4/3 fy = n v u + v. (3.3) h 4/3 The two components of the turbulent term S t have the following expressions S t = ( ν t h u ) + ( [ v ν t h x x y x + u ]), (3.3) y S t = ( [ v ν t h x x + u ]) + ( ν t h v ). (3.3) y y y Expressions (3.5)-(3.8) are known as the Shallow Water Equations in conservative form. It is common not to take into account neither the Coriolis and wind stress terms nor the turbulent term.

32 Chapter 4 Application to the D SWE 4. Types of finite volumes The use of the finite volume method in computational fluid dynamics is relatively recent. According to Hirsch [7, pg. 37] it was introduced by McDonald in 97 and independently by Mc Cormack and Paullay in 97 for solving the Euler equations in D. In 973 Rizzi and Inouye extended it to three-dimensional flows. Eymard et al. [3] attribute their introduction, ten years earlier, to Tichonov and Samarskii for solving convection-diffusion equations. In [3, pg. 9] a long list of works devoted to the mathematical analysis of the method is mentioned. To use this method we usually start from a previous discretization of the computational domain in elements, normally triangles or quadrangles, from which the new mesh or finite volume cells is built. In each of these volumes the integral form of the equations is obtained, simplified by applying the divergence theorem and discretized. The resulting expressions establish the exact conservation of relevant flow properties in each cell. The terms of the equations are replaced by approximations of the finite difference type, obtaining algebraic equations that are solved by an iterative process. We briefly describe below some of the most commonly used finite volumes [4, 5]. 4.. Cell-type finite volumes The cell type (or cell-center) finite volumes [5, pg. 366] are the same initial grid cells and the values of the dependent variables are stored in the cell center (centers of quadrilaterals or barycenters of triangles). This method has the advantage of using the initial mesh and the disadvantage that the nodes to which the variables values are assigned -which represent the cell 3

33 CHAPTER 4. APPLICATION TO THE D SWE 3 values and are used in the discretization- do not coincide with the nodes of the original mesh. 4.. Vertex-type finite volumes The vertex type (or cell-vertex) finite volumes [5, pg. 368] use the vertices of the initial mesh as nodes of the finite volume mesh and the new cells are built around them. In contrast to the previous case, the initial mesh vertices are used and assigned the variables values in each finite volume. In this method the implementation of boundary conditions is simpler, since the value of the variables in the boundary nodes are known. The drawback is that a new mesh has to be built (dual mesh) Edge-type finite volumes This type of finite volumes, not usual in literature, is described in [3, pg. 87]. To obtain them we start from a mesh formed by triangles, each of which is divided into three, by joining each vertex with the barycentre. Then the subtriangles are joined in pairs so that each finite volume is formed by the two subtriangles with an edge of the initial mesh in common. The center of the finite volume is the midpoint of the edge. With this method, the angular points of the contour -that belong to the initial mesh- are not nodes of the finite volume mesh, what avoids two difficulties. The first is related to the velocity vector: since fluid do not passes through the walls, in nodes corresponding to solid boundary the velocity vector must be parallel to the boundary, which gives problems in angular points. Another difficulty is the calculation of the perpendicular to the boundary edges at such points, which have two perpendicular. On the other hand, the initial nodes are not used. To obtain the values in these nodes we must interpolate. 4. Description of the finite volumes used The finite volumes used in this work are based on a triangular discretization of the domains so that the nodes of the triangular mesh are used as the nodes of the finite volume mesh. For each node I, the barycenters of the triangles with I as a common vertex and the mid-points of the edges that meet at I are taken. The boundary Γ i of the cell C i is obtained by joining these points and Γ ij = AMB represents the common part of Γ i and Γ j.